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# Mixed Number and Fraction Estimation

## Estimating sums and differences

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Mixed Number and Fraction Estimation

Remember the seventh grade bake sale from the Equivalent Fractions Concept? Well, look at this situation.

Sam baked $8 \frac{9}{12}$ batches of cookies for the bake sale. When he brought them to school, Tracy asked how many batches he had made.

When Sam told her, she wrote 9 batches of cookies on the check sheet.

How did Tracy know that Sam's quantity was close to 9 batches? Do you know?

This Concept will teach you how to approximate mixed numbers and fractions using benchmarks. By the end of it, you will know how Tracy came to this conclusion.

### Guidance

Because a whole can be divided into an infinite number of parts, it is sometimes difficult to get a good sense of the value of a fraction or mixed number when the denominator of the fraction is large. In order to get an approximate sense of the value of a fraction, we compare the complicated fraction with several simpler fractions, or benchmarks. The three basic fraction benchmarks are: 0, $\frac{1}{2}$ and 1.

When approximating the value of a fraction or mixed number, ask yourself which of these benchmarks is the number closest to?

Let’s look at how to apply benchmarks.

What is the approximate size of $\frac{17}{18}$ ?

To begin with, we need to determine whether the fraction is closest to 0, one-half or 1 whole. The denominator is 18 and the numerator is 17. The numerator is close in value to the denominator. The value of $\frac{17}{18}$ is closest to 1 because $\frac{18}{18}$ would be equal to one.

That’s right. When you are looking for a benchmark, you want to choose the one that makes the most sense.

What is the benchmark for $\frac{24}{49}$ ?

First, we can look at the relationship between the numerator and the denominator. The numerator in this case is almost half the denominator. Therefore the correct benchmark is one-half.

We can identify benchmarks for mixed numbers too. The difference is that rather than zero, we look to the whole number of the mixed number, the half and the whole number next in consecutive order.

What is the benchmark for $7 \frac{1}{8}$ ?

Here we have 7 and one-eighth. Is this closer to 7, $7 \frac{1}{2}$ or 8? If you think about it logically, one-eighth is a very small fraction. There is only one part out of eight. Therefore, it makes sense for our benchmark to be 7.

Choose the correct benchmark for each example.

#### Example A

$\frac{1}{12}$

Solution: 0

#### Example B

$\frac{5}{6}$

Solution: 1

#### Example C

$9 \frac{3}{9}$

Solution: 9

Here is the original problem once again.

Sam baked $8 \frac{9}{12}$ batches of cookies for the bake sale. When he brought them to school, Tracy asked how many batches he had made.

When Sam told her, she wrote 9 batches of cookies on the check sheet.

How did Tracy know that Sam's quantity was close to 9 batches? Do you know?

To figure out Tracy's decision, let's look at the fraction part of the mixed number of batches.

$8 \frac{9}{12}$

9 is more than half of 12, so rounded up to 9 batches. If the fraction part of 12 would have been less than half, then Tracy would have rounded down to 8 batches.

### Guided Practice

Here is one for you to try on your own.

Name the common benchmark for this fraction.

$\frac{4}{7}$

To begin, we have to look at the relationship between 4 and 7. 4 is a little more than half of seven. Because of this, we can say that this fraction is closest to one - half.

$\frac{1}{2}$ is the appropriate benchmark.

### Explore More

Directions: Approximate the value of the following fractions using the benchmarks 0, $\frac{1}{2}$ and 1.

1. $\frac{9}{10}$

2. $\frac{11}{20}$

3. $\frac{2}{32}$

4. $\frac{21}{22}$

5. $\frac{1}{23}$

6. $\frac{11}{100}$

7. $\frac{2}{3}$

8. $\frac{14}{28}$

9. $\frac{16}{30}$

10. $\frac{18}{21}$

Directions: Approximate the value of the following mixed numbers.

11. $2 \frac{79}{80}$

12. $6 \frac{1}{10}$

13. $43 \frac{7}{15}$

14. $8 \frac{7}{99}$

15. $6 \frac{21}{22}$

### Vocabulary Language: English

Denominator

Denominator

The denominator of a fraction (rational number) is the number on the bottom and indicates the total number of equal parts in the whole or the group. $\frac{5}{8}$ has denominator $8$.
Equivalent

Equivalent

Equivalent means equal in value or meaning.
Equivalent Fractions

Equivalent Fractions

Equivalent fractions are fractions that can each be simplified to the same fraction. An equivalent fraction is created by multiplying both the numerator and denominator of the original fraction by the same number.
fraction

fraction

A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a rational number.
Greatest Common Factor

Greatest Common Factor

The greatest common factor of two numbers is the greatest number that both of the original numbers can be divided by evenly.
improper fraction

improper fraction

An improper fraction is a fraction in which the absolute value of the numerator is greater than the absolute value of the denominator.
Mixed Number

Mixed Number

A mixed number is a number made up of a whole number and a fraction, such as $4\frac{3}{5}$.
Numerator

Numerator

The numerator is the number above the fraction bar in a fraction.
Simplify

Simplify

To simplify means to rewrite an expression to make it as "simple" as possible. You can simplify by removing parentheses, combining like terms, or reducing fractions.
Whole Numbers

Whole Numbers

The whole numbers are all positive counting numbers and zero. The whole numbers are 0, 1, 2, 3, ...